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1999 | 82 | 2 | 277-286
Tytuł artykułu

Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
ℒ denotes the Lebesgue measurable subsets of ℝ and $ℒ_0$ denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ\$ℒ_0$ has a perfect subset Q ∈ ℒ\$ℒ_0$ which is a subset of or misses M (a similar statement omitting "is a subset of or" characterizes $ℒ_0$). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the "Marczewski measurable sets" and the σ-ideal $(s^0)$ which we call the "Marczewski null sets". M ∈ (s) if every perfect set P has a perfect subset Q which is a subset of or misses M. M ∈ $(s^0)$ if every perfect set P has a perfect subset Q which misses M. In this paper, it is shown that there is a collection G of $G_δ$ sets which can be used to give similar "Marczewski-Burstin-like" characterizations of the collections $B_w$ (sets with the Baire property in the wide sense) and FC (first category sets). It is shown that no collection of $F_σ$ sets can be used for this purpose. It is then shown that no collection of Borel sets can be used in a similar way to provide Marczewski-Burstin-like characterizations of $B_r$ (sets with the Baire property in the restricted sense) and AFC (always first category sets). The same is true for U (universally measurable sets) and $U_0$ (universal null sets). Marczewski-Burstin-like characterizations of the classes of measurable functions are also discussed.
Rocznik
Tom
82
Numer
2
Strony
277-286
Opis fizyczny
Daty
wydano
1999
otrzymano
1999-03-22
poprawiono
1999-07-15
Twórcy
  • Department of Mathematics, Auburn University, Auburn, AL 36849-5310, U.S.A.
  • Department of Mathematics, Tuskegee University, Tuskegee, AL 36088, U.S.A.
Bibliografia
  • [1] M. Balcerzak, A. Bartoszewicz, J. Rzepecka and S. Wroński, Marczewski fields and ideals, preprint.
  • [2] S. Baldwin and J. Brown, A simple proof that $(s)/(s^0)$ is a complete Boolean algebra, Real Anal. Exchange, to appear.
  • [3] C. Burstin, Eigenschaften messbarer und nicht messbarer Mengen, Sitzungsber. Kaiserlichen Akad. Wiss. Math. Natur. Kl. Abt. IIa 123 (1914), 1525-1551.
  • [4] C. Kuratowski, La propriété de Baire dans les espaces métriques, Fund. Math. 16 (1930), 390-394.
  • [5] E. Marczewski (Szpilrajn), Sur une classe de fonctions de M. Sierpiński et la classe correspondante d'ensembles, ibid. 24 (1935), 17-34.
  • [6] E. Marczewski (Szpilrajn), Sur les ensembles et les fonctions absolument mesurables, C. R. Soc. Sci. Varsovie 30 (1937), 39-68.
  • [7] J. Morgan II, Measurability and the abstract Baire property, Rend. Circ. Mat. Palermo (2) 34 (1985), 234-244.
  • [8] J. Morgan II, Point Set Theory, Marcel Dekker, New York and Basel, 1990.
  • [9] O. Nikodym, Sur la condition de Baire, Bull. Internat. Acad. Polon. 1929, 591-598.
  • [10] P. Reardon, Ramsey, Lebesgue, and Marczewski sets and the Baire property, Fund. Math. 149 (1996), 191-203.
  • [11] M. Ruziewicz, Sur une propriété générale des fonctions, Mathematica (Cluj) 9 (1935), 83-85.
  • [12] W. Sierpiński, Sur un problème de M. Ruziewicz concernant les superpositions de fonctions jouissant de la propriété de Baire, Fund. Math. 24 (1935), 12-16.
  • [13] J. Walsh, Marczewski sets, measure and the Baire property, II, Proc. Amer. Math. Soc. 106 (1989), 1027-1030.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-cmv82i2p277bwm
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